Dimensionality Reduction

Objectives

Code Implementation

Code

import pandas as pd
cities = pd.read_csv('../data/cleaned_data/apta_cities_cleaned.csv')
cities = cities.drop(columns=['Unnamed: 0'])
cities.head()

Intel MKL WARNING: Support of Intel(R) Streaming SIMD Extensions 4.2 (Intel(R) SSE4.2) enabled only processors has been deprecated. Intel oneAPI Math Kernel Library 2025.0 will require Intel(R) Advanced Vector Extensions (Intel(R) AVX) instructions.
Intel MKL WARNING: Support of Intel(R) Streaming SIMD Extensions 4.2 (Intel(R) SSE4.2) enabled only processors has been deprecated. Intel oneAPI Math Kernel Library 2025.0 will require Intel(R) Advanced Vector Extensions (Intel(R) AVX) instructions.

	City	Population	Area	Cost_per_trip	Fare_per_trip	Miles_per_trip
0	Seattle--Tacoma, WA	3544011	982.52	13.906032	1.570667	5.786344
1	Spokane, WA	447279	171.67	13.433827	0.988308	4.772569
2	Yakima, WA	133145	55.77	19.720093	1.112531	5.179168
3	Eugene, OR	270179	73.49	10.851494	2.753356	3.684118
4	Portland, OR--WA	2104238	519.30	10.804361	1.025659	4.011388

Code

import json
import numpy as np
import matplotlib.pyplot as plt
from sklearn.metrics import silhouette_samples, silhouette_score

X = cities.drop(columns=['City']).to_numpy()

print('NUMERIC MEAN:\n',np.mean(X,axis=0))
print("X SHAPE",X.shape)
print("NUMERIC COV:")
print(np.cov(X.T))

from numpy import linalg as LA
w, v1 = LA.eig(np.cov(X.T))
print("\nCOV EIGENVALUES:",w)
print("COV EIGENVECTORS (across rows):")
print(v1.T)

NUMERIC MEAN:
 [7.63817374e+05 2.54954371e+02 1.62164796e+01 1.69764181e+00
 6.02033451e+00]
X SHAPE (286, 5)
NUMERIC COV:
Intel MKL WARNING: Support of Intel(R) Streaming SIMD Extensions 4.2 (Intel(R) SSE4.2) enabled only processors has been deprecated. Intel oneAPI Math Kernel Library 2025.0 will require Intel(R) Advanced Vector Extensions (Intel(R) AVX) instructions.
[[ 3.00348049e+12  6.21976504e+08 -2.68509287e+06 -2.80714130e+05
  -3.74964286e+05]
 [ 6.21976504e+08  1.60662563e+05 -7.09356971e+02 -8.34065149e+01
  -1.18548286e+02]
 [-2.68509287e+06 -7.09356971e+02  1.12521769e+02  1.14344019e+01
   1.54309551e+01]
 [-2.80714130e+05 -8.34065149e+01  1.14344019e+01  1.07182685e+01
   6.17463373e+00]
 [-3.74964286e+05 -1.18548286e+02  1.54309551e+01  6.17463373e+00
   2.61947556e+01]]
Intel MKL WARNING: Support of Intel(R) Streaming SIMD Extensions 4.2 (Intel(R) SSE4.2) enabled only processors has been deprecated. Intel oneAPI Math Kernel Library 2025.0 will require Intel(R) Advanced Vector Extensions (Intel(R) AVX) instructions.

COV EIGENVALUES: [3.00348062e+12 3.18612161e+04 1.13357226e+02 2.46043548e+01
 8.18652852e+00]
COV EIGENVECTORS (across rows):
[[-9.99999979e-01 -2.07085246e-04  8.93993769e-07  9.34629441e-08
   1.24843257e-07]
 [ 2.07087147e-04 -9.99987172e-01  4.82944608e-03  7.95476612e-04
   1.28713329e-03]
 [-1.36812764e-07  5.03927972e-03  9.77725251e-01  1.15731143e-01
   1.75026404e-01]
 [ 1.31093957e-07 -4.57011124e-04  1.99629231e-01 -2.56090918e-01
  -9.45814677e-01]
 [ 2.27827260e-08 -9.92273370e-05  6.46388484e-02 -9.59699490e-01
   2.73493506e-01]]

Code

from sklearn.preprocessing import StandardScaler

X = StandardScaler().fit_transform(X)
cov = (X.T @ X) / (X.shape[0] - 1)
w, v1 = np.linalg.eig(cov)

idx = np.argsort(w, axis=0)[::-1]
sorted_eig_vectors = v1[:, idx]

cumsum = np.cumsum(w[idx]) / np.sum(w[idx])
xint = range(1, len(cumsum) + 1)
plt.plot(xint, cumsum)

plt.xlabel("Number of components")
plt.ylabel("Cumulative explained variance")
plt.xticks(xint)
plt.xlim(1, 5, 1)

Intel MKL WARNING: Support of Intel(R) Streaming SIMD Extensions 4.2 (Intel(R) SSE4.2) enabled only processors has been deprecated. Intel oneAPI Math Kernel Library 2025.0 will require Intel(R) Advanced Vector Extensions (Intel(R) AVX) instructions.

/var/folders/z5/l6g0391s0qg3vsbnvl7y81n80000gn/T/ipykernel_36075/3215643524.py:17: MatplotlibDeprecationWarning: Passing the emit parameter of set_xlim() positionally is deprecated since Matplotlib 3.6; the parameter will become keyword-only two minor releases later.
  plt.xlim(1, 5, 1)

(1.0, 5.0)

Code

import seaborn as sns
X_df = pd.DataFrame(X).rename(columns={0: 'Population', 1: 'Area', 2: 'Cost_per_trip', 3: 'Fare_per_trip', 4: 'Miles_per_trip'})
sns.heatmap(X_df.corr())

<Axes: >

Code

from sklearn.decomposition import PCA
pca = PCA(n_components=4)
pca.fit(X)
data_pca = pca.transform(X)
data_pca = pd.DataFrame(data_pca,columns=['PC1','PC2','PC3','PC4'])
data_pca.head()

	PC1	PC2	PC3	PC4
0	2.195331	1.041785	-0.110497	-0.016923
1	-0.031348	-0.494236	0.066595	0.048293
2	-0.552852	-0.343962	-0.319175	-0.122077
3	-0.252383	-0.531766	0.230998	0.663955
4	1.206063	-0.041013	0.042951	0.169983

Code

sns.heatmap(data_pca.corr())

<Axes: >

Code

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
%matplotlib widget
fig = plt.figure()
ax = Axes3D(fig)
ax.scatter(data_pca['PC2'],data_pca['PC3'],data_pca['PC4'], c=data_pca['PC1'])
ax.set_title("3D Plot of Principal Components")
ax.set_xlabel('PC2')
ax.set_ylabel('PC3')
ax.set_zlabel('PC4')
plt.show()

Code

from sklearn.manifold import TSNE
X_embedded = TSNE(n_components=3, learning_rate='auto',init='random', perplexity=1).fit_transform(X)

# EXPLORE RESULTS
print("RESULTS") 
print("shape : ",X_embedded.shape)
print("First few points : \n",X_embedded[0:4,:])

# PLOT 
plt.scatter(X_embedded[:,0],X_embedded[:,1], alpha=0.5)
plt.show()

RESULTS
shape :  (286, 3)
First few points : 
 [[ -9.726924   -16.770332   -11.739644  ]
 [  1.2530551  -14.637746    -0.95117176]
 [ 28.71916      1.3856144   -0.31036332]
 [  5.873687     4.3193254   21.88686   ]]

Code

X_embedded = TSNE(n_components=3, learning_rate='auto',init='random', perplexity=4).fit_transform(X)

# EXPLORE RESULTS
print("RESULTS") 
print("shape : ",X_embedded.shape)
print("First few points : \n",X_embedded[0:4,:])

# PLOT 
plt.scatter(X_embedded[:,0],X_embedded[:,1], alpha=0.5)
plt.show()

RESULTS
shape :  (286, 3)
First few points : 
 [[ 10.331259  -10.506528  -11.510693 ]
 [ -6.15948    -1.0071286   2.6866064]
 [  1.8365389   5.3667107  10.372743 ]
 [-12.317913   -8.945713    2.5623868]]

Reporting